Square Root of 728

The square root is the inverse of the square of a number. 728 is not a perfect square. The square root of 728 can be expressed in both radical and exponential forms. In radical form, it is expressed as √728, whereas in exponential form it is (728)^(1/2). √728 ≈ 26.9737, which is an irrational number because it cannot be expressed as a ratio of two integers (p/q, where q ≠ 0).

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, methods like the long division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 728 is broken down into its prime factors.

Step 1: Finding the prime factors of 728

Breaking it down, we get 2 × 2 × 2 × 91 = 2^3 × 7 × 13

Step 2: Now we found the prime factors of 728. The second step is to make pairs of those prime factors. Since 728 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √728 using prime factorization alone is impractical.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 728, we need to group it as 28 and 7.

Step 2: Now we need to find n whose square is less than or equal to the first group (7). We can say n is ‘2’ because 2 × 2 = 4, which is less than or equal to 7. Now the quotient is 2; after subtracting 4 from 7, the remainder is 3.

Step 3: Now, bring down 28 to make it 328, which is the new dividend. Double the current quotient (2) to get 4, which becomes our new divisor (4_).

Step 4: Find a digit to fill in the blank (4_) such that the product of this new number and the same digit is less than or equal to 328.

Step 5: Continue this process until you achieve the desired precision. Using this method, you will find that √728 ≈ 26.97.

The approximation method is another method for finding the square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 728 using the approximation method.

Step 1: Find the closest perfect squares around 728. The smallest perfect square less than 728 is 676 (26^2), and the largest perfect square greater than 728 is 729 (27^2). Thus, √728 falls between 26 and 27.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (728 - 676) / (729 - 676) = 52/53 ≈ 0.9811 Add this decimal to the smaller square root: 26 + 0.9811 = 26.9811 Therefore, the square root of 728 is approximately 26.9811.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Long division method: A mathematical procedure used to find the square root of numbers, especially non-perfect squares, by breaking down the calculation into simpler steps.

• Approximation method: A technique to estimate the value of a square root by using nearby perfect squares to find a closer value.

• Prime factorization: The process of expressing a number as a product of its prime factors.